Title: The Uhlmann Phase Winding in Bose-Einstein Condensates at Finite Temperature

Abstract: We investigate the Uhlmann phase, a generalization of the celebrated Berry phase, for Bose-Einstein condensates (BECs) at finite temperature. The Uhlmann phase characterizes topological properties of mixed states, in contrast to the Berry phase which is defined for pure states at zero temperature. Using the $SU(1,1)$ symmetry of the Bogoliubov Hamiltonian, we derive a general formula for the Uhlmann phase of BECs. Numerical calculations reveal that the Uhlmann phase can differ from the Berry phase in the zero-temperature limit, contrary to previous studies. As the temperature increases, the Uhlmann phase exhibits a winding behavior, and we relate the total winding degree to the Berry phase. This winding indicates that the Uhlmann phase takes values on a Riemann surface. Furthermore, we propose an experimental scheme to measure the Uhlmann phase of BECs by purifying the density matrix using an atomic interferometer.